# EUREKA!!!!!!!!!! (ha ha ha ha ha ha) >:-)

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On Mon, 13 Nov 1995, Steve VanDevender wrote:
> Regarding this whole momentum-transfer issue, in some far-distant
> posting to the newsletter I pointed out that a reflective sail is
> the most efficient option for accelerating a spacecraft with
> beamed power; it changes photons with momentum +p into photons
> with momentum -p, and thereby increases the spacecraft momentum
> by 2p.  It is unlikely that you could absorb the photons and use
> them to power a mass driver with greater efficiency than the
> reflectivity of a mirror.
>

Agreed.

> Using beamed power to decelerate a ship that's going in the same
> direction as the photons is more problematic.  At best the
> efficiency is going to be bad.
>
It is, which is why I proposed a linear accelerator in the first place.

I HAVE SOLVED THE PROBLEM OF SLOWING DOWN!!!! <smug grin>

(Okay, Okay, I'll calm down now.  And promise not to be so smug for the
rest of this message)

To Review:
The main problem is that the momentum of the recieved photons is greater
than the emitted exhaust.  This is as it should be, the exhaust can't
travel at C, and besides there's always losses.  Solar electric space
craft work because the ejected ion stream is traveling at far less than
the speed of light, so the energy needed is less.  and besides, the light
is almost never coming from straight behind.

But I digress.

First, there is _no_ way to reduce the momentum imparted by the photons.
However, we _can_ change the _direction_ of the thrust.  or to paraphrase
an old Chinese proverb.  You can't change the direction of the wind, but
you can adjust the sail.  Do you see it yet? the solution?

Tilted sail.

Physical Design. (highly stylized ascii art)

//////////      \\\\\\\\\\  <---- Sail elements

|~~|
|  |  <- ship    Note: support cables omitted
|  |
|  |
y           |  |
^           |  |      Energy is beamed from below in this
|           |  |      diagram.
|--> x      |  |
|__|

The Sail (and it is a sail for most of the flight) is made of concentric
conic sections tilted at 76.6 degrees (more on this later) this changes
the direction of the imparted velocity, and since the sail elements have
a circular symetry, the x componets of the force cancel out.

A Force Balance on one sail element:

Velocity  y          /  <-- Sail element
Vector\  component  /
\     |     /
\    |    /
\   |   /
\  |  /
\ | /
x component_________\|/
/   angle  T (for theta)
/
/  ________________________

There is another element on the other side of the ship whose x component
cancels this one.  The y component is Fy=F*cos (T) where F equals E/c.

Some constants:
c  - Speed of Light (of course)	         = 3.0 E+08 m/s
Ms - Mass of ship (RM neglected for now) = 2.5 E+09 Kg
G  - Ship's Gravity                      = 10 m/s^2
Ve - Exhaust velocity                    = .9996 c
Eb - Beamed energy                       = 3.24 E+19 Watts
Pe - Engine Thrust                       = 2 G
T  - angle of sail elements              = 76.6 degrees
(this value was found by trial)
(and error and is critical    )

Some equations:

Photon Thrust(Newtons)    Pp = E / c * cos (T) / Ms
Pp=3.24E+19 / 3.0 E+08 * cos (76.6) / 2.5 E+09 = 10 m/s^2

Exhaust Mass (Kg/sec)     Me = Pe * Ms / (Ve * c)
Me = 20 * 2.5 E+09 / (0.9996 * 3 E+08) = 166.73 Kg/sec

Exhaust Rest Mass (Kg/sec)Mr = Me * SQRT(1- Ve^2)
Mr = 166.73 * Sqrt (1-0.9996^2) = 4.72 Kg/sec

Kinetic Energy in Exhaust Ke = 1/2 * Mr * (Ve *c)^2
Ke = 1/2 * 4.72 * (0.9996 * 3 E+08) = 2.12 E+17

Relativistic Energy       Er = (Me-Mr)*c^2
Er = (166.73 - 4.72) * 3 E+08^2 = 1.46 E+19

Total energy needed       Et = Er + Ke
Et = 2.12 E+17 + 1.46 E+19 = 1.48 E+19

Energy Available          Ea = Eb * .8 (conversion losses)
Ea = 3.24 E+19 * .8 = 1.66 E+19

And as you can see, the energy Available exceeds the energy needed.  No
physical laws get broken.  During the Accel phase, the engines are
quiet.  The "Asimov" accelerates by reflected maser power  (the
transmitters operate at 1/2 power because the "Asimov" is Reflecting the
maser, not Absorbing it like the above equations assume.

This solution is more expensive (in terms of energy and sail area)

it has a few extra benefits:
the radial force component (x in the above force diagram) allows us to use
tensile forces to hold the sail together  (yes, it's a lot of force, but
I think it is possible to do from an engineering standpoint)

Kevin

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